The components of the background of a Ge(Li) detector and an ultrapure Ge detector in gamma spectrometers in passive shielding with a special design were studied in a ground-surface laboratory in 1996-2006. The measurement time varied from 45 to 240 h. The background for the detectors is due to radionuclides in the shielding material and cavities in the shielding and the detector materials themselves. Special attention is devoted to the study of the time dependence found for the background of the daughter of products of 222 Rn decay, including the 46.5 keV peak of 210 Pb.